\vspace{-0.1in}
\section{Conclusion}
\vspace{-0.1in}
We studied a generalization of the random walk, namely the cobra walk,
and analyzed its cover time for trees, grids, and expander graphs.
The cobra walk is a natural random process, with potential
applications to epidemics and gossip-based information spreading.  We
plan to explore further the connections between cobra walks and the
SIS model, and pursue their practical implications.  From a
theoretical standpoint, there are several interesting open problems
regarding cobra walks that remain to be solved.  First is to obtain a
tight bound for the cover time of cobra walks on expanders.  Our upper
bound is $O(\log^2 n)$, while the diameter $\Omega(\log n)$ is a basic
lower bound.  Another pressing open problem is to determine the
worst-case bound on the cover time of cobra walks on general graphs.
It will also be interesting to establish and compare the message
complexity of cobra walk with the standard random walk and other
gossip-based rumor spreading processes.

\junk{
Random
walks have extensive applications in networks, we hope that cobra
walks will also be useful, with the additional property of faster
coverage. 



In general, unlike the standard random
walk and other gossip-based protocols like push which have a
well-developed theory, we know little about the properties of cobra
walks in general graphs. For example, what is the worst case cover
time of a cobra walk, and how does it vary with the branching factor
$k$? It is clear that the cover time is not worse than a standard
random walk, but it will be interesting to establish tight asymptotic
bounds.  The techniques
used here, might be helpful in further analysis of cobra walks.
}
